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Toroidal compactificationsof discrete quotients of period domains

Azniv Kasparian 1

1Partially supported by Contract 144/ 2015 with the the ScientificFoundation of Kliment Ohridski University of Sofia.

Toroidal compactifications of period domains

Algebraic family of projective algebraic varieties

Let X ⊂ PN(C) be a smooth quasi-projective variety,

[ω] be the Kahler class of the Fubini-Study metric on PN(C) and

π : X→ S be a proper morphism onto a quasi-projective varietyS with a smooth compactification S by a divisor S \ S withnormal crossings. S is called a moduli space.

The fibres Xs := π−1(s) are mutually diffeomorphic, connectedsmooth projective varieties of dimCXs = n.

The intersection form on the primitive cohomologies

The primitive cohomologiesHn

prim(Xs,Q) := ker[∧[ω] : Hn(Xs,Q)→ Hn+2(Xs,Q)]determine a lattice Hn

Z := Hn(Xs,Z) ∩Hnprim(Xs,Q),

endowed with a non-degenerate bilinear intersection formQ : Hn

Z ×HnZ → Z, Q([ω1], [ω2]) =

√−1n(n−1) ∫

Xsω1 ∧ ω2.

For an odd weight n = 2m + 1 the form Q is skew-symmetric,while for an even n = 2m the form Q is symmetric.

Z ×HnZ → Z, Q([ω1], [ω2]) =

√−1n(n−1) ∫

Xsω1 ∧ ω2.

Z ×HnZ → Z, Q([ω1], [ω2]) =

√−1n(n−1) ∫

Xsω1 ∧ ω2.

Hodge-Riemann bilinear relations

By Dolbeault’s Theorem, HnC := Hn

Z ⊗Z C admits a Hodgedecompositions Hn

C = ⊕nj=0H

n−j,j(Xs) for all s ∈ S.

The first Hodge-Riemann bilinear relation asserts thatQ(ϕ,ψ) = 0 for ∀ϕ ∈ Hn−j,j, ∀ψ ∈ Hn−i,i with i 6= n− j.

The second Hodge-Riemann bilinear relation states that√−1n−2jQ(ϕ,ϕ) > 0 for ∀ϕ ∈ Hn

C = ⊕nj=0H

The period domain D

The variation of the complex structure on the fibres Xs = π−1(s)of π : X→ S induces a variation of the Hodge decompositions

HnC = Hn

prim(Xs,C) = ⊕nj=0H

n−j,j(Xs).

The classifying space D of the Hodge decompositionsHn

C = HnZ ⊗Z C = ⊕n

j=0Hn−j,j with fixed lattice Hn

Z, polarizationQ : Hn

Z ×HnZ → Z and Hodge numbers hn−j,j = dimCHn−j,j is

called a period domain.

The linear algebraic group GR = Sp(HnR,Q) for n = 2m + 1 or

GR = SO(HnR,Q) for n = 2m acts transitively on the complex

manifold D with compact stabilizer V of the origin o ∈ D.

The period domain D

HnC = Hn

n−j,j(Xs).

The period domain D

HnC = Hn

n−j,j(Xs).

The compact dual D of D

Let D be the classifying space of the decreasing filtrationsF• = {F0 ⊃ F1 ⊃ . . . ⊃ Fn}, Fk = ⊕n

s=kHs,n−s, subject to the

first Hodge Riemann bilinear relation Q(Fk,Fn−k+1) = 0.

As a closed subvariety of a product of Grassmannians, D is acompact homogeneous space D = GC/PC of the algebraic groupGC := {g ∈ SL(Hn

C) |Q(gu, gv) = Q(u, v), ∀u, v ∈ HnC}.

The period domain D is an open complex submanifold of D andD is called the compact dual of D.

C) |Q(gu, gv) = Q(u, v), ∀u, v ∈ HnC}.

Period map

The correspondence, associating to Xs = π−1(s) the Hodgedecomposition Hn

C = ⊕nj=0H

n−j,jprim (Xs) provides a holomorphic

period map Φ : S→ D/Γ for a lattice Γ ≤ GZ = SLZ(HnZ,Q).

Any period map Φ : S→ D/Γ admits a holomorphic liftingΦ : S→ D to the universal covering S of S.

If so ∈ S and Φ(so) = o ∈ D then the differential(dΦ)soT

1,0so S→ T1,0

o D ⊂ g := LieGC takes values in thehorizontal subspace g−1,1 = {X ∈ g |X(Hn−j,j

o ) ⊆ Hn−j−1,j+1o }.

Period map

C = ⊕nj=0H

1,0so S→ T1,0

o ) ⊆ Hn−j−1,j+1o }.

Period map

C = ⊕nj=0H

1,0so S→ T1,0

o ) ⊆ Hn−j−1,j+1o }.

The aim of the talk

Problem: To construct a complex analytic compactification(D/Γ)Σ of D/Γ, such that any period map Φ : S→ D/Γ has aholomorphic extension Φ : S→ (D/Γ)Σ to a smoothcompactification S of S.

Motivation for the study of the problem: To describe thedegeneration of the Hodge structure on the primitivecohomologies Hn

prim(Xs,C) of a smooth projective variety Xs,when Xs is deformed to a singular birational model.

The aim of the talk

Problem: To construct a complex analytic compactification(D/Γ)Σ of D/Γ, such that any period map Φ : S→ D/Γ has aholomorphic extension Φ : S→ (D/Γ)Σ to a smoothcompactification S of S.

Motivation for the study of the problem: To describe thedegeneration of the Hodge structure on the primitivecohomologies Hn

prim(Xs,C) of a smooth projective variety Xs,when Xs is deformed to a singular birational model.

Γ-rational parabolic subgroups

A subgroup P of a linear algebraic group G is parabolic ifcontains a maximal connected solvable algebraic subgroup of G.

For any lattice Γ of G, a parabolic subgroup P of G is Γ-rationalif NP ∩ Γ is a lattice of the unipotent radical NP of P.

Let us denote by MParΓ the set of the Γ-rational maximalparabolic subgroups of G.

Classical toroidal compactificationsof local Hermitian symmetric spaces

If D = GR/K is a Hermitian symmetric space of non-compacttype then P ∈ MParΓ are exactly the normalizers of theΓ-rational analytic boundary components D(P) of D.

A monograph of Ash, Mumford, Rapoport and Tai from 1975builds up complex analytic toroidal compactifications (D/Γ)Σ ofthe non-compact quotients D/Γ by lattices Γ ≤ GZ.

For any P ∈ MParΓ, let us consider the latticeΥP = Γ ∩UP ' (Zm,+) of the center UP ' (Rm,+) of theunipotent radical NP of P.

exp(2πiuj) : (Cm,+) ' UP ⊗R C→ (UP ⊗R C)/ΥP =: T(P)maps onto a complex algebraic torus T(P) ' (C∗)m.

There is a diffeomorphic Siegel domain splittingD/ΥP ' (UP + iCP)/ΥP × (NP/UP)×D(P)

for an open homogeneous cone cone CP ⊂ UP, a complex vectorspace NP/UP and a bounded symmetric domain D(P).

Let XΣ(P) ⊃ T(P) be the toric variety, associated with anappropriate family Σ(P) of polyhedral cones in UP andYΣ(P) be the interior of the closure of (UP + iCP)/ΥP in XΣ(P).

The partial compactification(D/ΥP)Σ(P) := YΣ(P) × (NP/UP)×D(P).

For any P1,P2 ∈ MParΓ with D(P1) ⊂ D(P2) there is aholomorphic map µP2

P1: (D/ΥP2)Σ(P2) → (D/ΥP1)Σ(P1).

The toroidal compactification (D/Γ)Σ, Σ = {Σ(P)}P∈MParΓis

obtained from∐

P∈MParΓ(D/ΥP)Σ(P) by gluing along the

holomorphic maps µP2P1

for D(P1) ⊂ D(P2) and the Γ-orbits.

P1: (D/ΥP2)Σ(P2) → (D/ΥP1)Σ(P1).

obtained from∐

P1: (D/ΥP2)Σ(P2) → (D/ΥP1)Σ(P1).

obtained from∐

Logarithmic manifolds

A monoid (M, .) is a non-empty subset of an abelian group(G, .), which is closed under the group operation and containsthe neutral element of G.

A local logarithmic structure on a local complex analytic space(U,OX(U)) is a monoid (MX(U), .) ⊃ (OX(U)∗, .) endowed witha homomorphism of monoids (MX(U), .)→ (OX(U), .).

A logarithmic manifold (X,OX,MX) is a complex analyticmanifold (X,OX) with a sheafMX of local logarithmicstructures.

Kato-Usui’s DΣ/Γ for a period domain D

Let D = GR/V be a period domain and Γ < GZ be a neatsubgroup, i.e., the subgroup of C∗, generated by the eigenvaluesof all the elements of Γ is torsion free.

In 2009 Kato and Usui construct Hausdorff spaces DΣ/Γ, whichare logarithmic manifolds.

Let D = GR/V be a period domain and Γ < GZ be a neatsubgroup, i.e., the subgroup of C∗, generated by the eigenvaluesof all the elements of Γ is torsion free.

In 2009 Kato and Usui construct Hausdorff spaces DΣ/Γ, whichare logarithmic manifolds.

DΣ depends on a Γ-invariant family Σ of nilpotent conesσ = R≥0N1 + . . .+ R≥0Nm ⊂ LieGQ with commuting elements.

The set DΣ consists of the pairs (σ,Z = exp (σC)F•) withσ ∈ Σ, Z ⊂ D, such that NFk ⊆ Fk−1 for ∀N ∈ σ, ∀k and

j=1ziNj

)F• ∈ D for all zj ∈ C with Imzj >> 0.

In particular, D ' {({0},F•) |F• ∈ D} ⊂ DΣ.

j=1ziNj

The boundary of a moduli space S

Let π : X→ S be an algebraic family of projective algebraicvarieties with a quasi-projective base S.

One can resolve the singularities of the projective closure S of Sin such a way that S \ S is a divisor with normal crossings.

Any point so ∈ S \ S has a neighborhood ∆d 'W(so) ⊂ S,d = dimC S in S, such that W(so) ∩ S ' (∆∗)k ×∆d−k.

The compact S = S ∪(∪so∈S\SW(so)

)has a finite sub-covering

S = S ∪W1 ∪ . . . ∪Wν .

The upper half plane H is an SL(2,R)-orbit

The upper half planeH := {ζ ∈ C | Imζ > 0} = SL(2,R)/SO(2)

is an orbit of the linear algebraic groupSL(2,R) := {A ∈ M2×2(R) | detA = 1}

with a compact isotropy groupSO(2) := {A ∈ SL(2,R) |AtA = I2}.

For any lattice Γ ≤ SL(2,Z), the Γ-rational parabolic subgroupsof SL(2,R) are the stabilizers of the Γ-rational boundary points

Q ∪ {i∞} = ∂ΓH.

The upper half plane H is an SL(2,R)-orbit

The upper half planeH := {ζ ∈ C | Imζ > 0} = SL(2,R)/SO(2)

is an orbit of the linear algebraic groupSL(2,R) := {A ∈ M2×2(R) | detA = 1}

with a compact isotropy groupSO(2) := {A ∈ SL(2,R) |AtA = I2}.

For any lattice Γ ≤ SL(2,Z), the Γ-rational parabolic subgroupsof SL(2,R) are the stabilizers of the Γ-rational boundary points

Q ∪ {i∞} = ∂ΓH.

The standard parabolic subgroup of SL(2,R)

All the Γ-rational parabolic subgroups of SL(2,R) are conjugateto the stabilizer Po of i∞ ∈ ∂ΓH in SL(2,R), which is called thestandard parabolic subgroup of SL(2,R).

There is Langlands decomposition Po ' NoAoZowith unipotent radical No = exp(RN) ' (R,+) of Po,

tangent to N =

(0 10 0

), split abelian component